On the derivative of the associated Legendre function of the first kind of integer order with respect to its degree

In our recent works [R. Szmytkowski, J. Phys. A 39 (2006) 15147; corrigendum: 40 (2007) 7819; addendum: 40 (2007) 14887], we have investigated the derivative of the Legendre function of the first kind, $P_{\nu}(z)$, with respect to its degree $\nu$. In the present work, we extend these studies and construct several representations of the derivative of the associated Legendre function of the first kind, $P_{\nu}^{\pm m}(z)$, with respect to the degree $\nu$, for $m\in\mathbb{N}$. At first, we establish several contour-integral representations of $\partial P_{\nu}^{\pm m}(z)/\partial\nu$. They are then used to derive Rodrigues-type formulas for $[\partial P_{\nu}^{\pm m}(z)/\partial\nu]_{\nu=n}$ with $n\in\mathbb{N}$. Next, some closed-form expressions for $[\partial P_{\nu}^{\pm m}(z)/\partial\nu]_{\nu=n}$ are obtained. These results are applied to find several representations, both explicit and of the Rodrigues type, for the associated Legendre function of the second kind of integer degree and order, $Q_{n}^{\pm m}(z)$; the explicit representations are suitable for use for numerical purposes in various regions of the complex $z$-plane. Finally, the derivatives $[\partial^{2}P_{\nu}^{m}(z)/\partial\nu^{2}]_{\nu=n}$, $[\partial Q_{\nu}^{m}(z)/\partial\nu]_{\nu=n}$ and $[\partial Q_{\nu}^{m}(z)/\partial\nu]_{\nu=-n-1}$, all with $m>n$, are evaluated in terms of $[\partial P_{\nu}^{-m}(\pm z)/\partial\nu]_{\nu=n}$.Comment: LateX, 40 pages, 1 figure, extensive referencin

3x+13x+1 inverse orbit generating functions almost always have natural boundaries

The $3x+k$ function $T_{k}(n)$ sends $n$ to $(3n+k)/2$ resp. $n/2,$ according as $n$ is odd, resp. even, where $k \equiv \pm 1~(\bmod \, 6)$. The map $T_k(\cdot)$ sends integers to integers, and for $m \ge 1$ let $n \rightarrow m$ mean that $m$ is in the forward orbit of $n$ under iteration of $T_k(\cdot).$ We consider the generating functions $f_{k,m}(z) = \sum_{n>0, n \rightarrow m} z^{n},$ which are holomorphic in the unit disk. We give sufficient conditions on $(k,m)$ for the functions $f_{k, m}(z)$ have the unit circle $\{|z|=1\}$ as a natural boundary to analytic continuation. For the $3x+1$ function these conditions hold for all $m \ge 1$ to show that $f_{1,m}(z)$ has the unit circle as a natural boundary except possibly for $m= 1, 2, 4$ and $8$. The $3x+1$ Conjecture is equivalent to the assertion that $f_{1, m}(z)$ is a rational function of $z$ for the remaining values $m=1,2, 4, 8$.Comment: 15 page

Metabolism and toxicity of two new benzodiazepine-type antileishmanial agents in hepatocytes and macrophages

With increasing reports of resistance of Leishmania to antimonials (Thakur et al., 2004) and other traditional antileishmanial drugs, the need for the discovery of new antileishmanial agents is rising. In an attempt to find new antileishmanial agents, two new benzodiazepine (BNZ) analogues (7-chloro-4-(cyclohexylmethyl)-1-methyl-3,4-dihydro-1H-1,4-benodiazepine-2,5-dione (BNZ-1) and 4-(cyclohexylmethyl)-1-methyl-3,4-dihydro-1H-1,4-benzodiazepine-2,5-dione (BNZ-2)) were synthesised, and found to be effective against leishmaniasis in mice. This study investigates the metabolism of these two drugs together with the prototype BNZ, flurazepam (FZP), using rat hepatocytes, and investigates their interaction with glutathione in macrophages. Hepatocytes (>80% viability by Trypan Blue exclusion isolated by liver perfusion with collagenase) were prepared from male Sprague-Dawley rats (200-250 g). Drugs (100 μM) were incubated with 2 × 106 viable cells/ml in Krebs-Hepes buffer, pH 7.4, in rotating round bottomed flasks under an atmosphere of 95% O2/5% CO2, at 37 °C for 3 h, and timed samples taken for metabolite measurement. Samples were extracted twice with ethyl acetate at pH 10, the combined organic phases evaporated to dryness and stored at −20 °C until analysis. Metabolites were separated by HPLC using an ACE C18 column (50 mm × 3.0 mm i.d., 5 μm packing), and a solvent gradient consisting of 0.1% formic acid: acetonitrile (starting composition 95:5%, and composition after 25 min 65:35% for FZP and 70:30% for both BNZ 1 and 2). Flow rate was 0.5 ml/min, and detection was at 230 nm. Identification of the metabolites was by mass spectrometry with both positive and negative ion electronspray ionization. The effects of 24 h exposure to the compounds (100 μM) was investigated in the macrophage cell line J774.1 in terms of reduced glutathione content (GSH) and the activity of glutathione reductase (GR). There was no evidence of significant cytotoxicity with any of the compounds at the concentration used. FZP (m/z 388) was metabolised by dealkylation of the two N-1 ethyl substituents (m/z 360 and m/z 332), followed by hydroxylation on the BNZ ring. There was no evidence for either N- or O-glucuronidation of the resulting metabolites. BNZ-1 (m/z 321) was metabolised by N-demethylation (m/z 307) followed by hydroxylation (m/z 323), mono- and di-hydroxylation of the parent (m/z 337 and m/z 353) and by glucuronidation of the mono-hydroxylated metabolite (m/z 513). BNZ-2 (m/z 287) was transformed by N-demethylation (m/z 273) and hydroxylation of the parent (m/z 303), with the latter further metabolised by O-glucuronidation (m/z 479). In addition, the hydroxylated N-demethylated product (m/z 289) was also formed. Macrophages did not produce detectable metabolism of any of the drugs. All the drugs depleted macrophage GSH significantly (p < 0.05 by ANOVA followed by Dunnett's test) with BNZ-1 and BNZ-2 causing greater depletion than FZP (40.6 ± 4.0 and 45.8 ± 8.4, respectively, compared with 55.5 ± 4.9 nmol/mg protein with FZP, n = 3). Control macrophage GSH was 74.1 ± 6.6 nmol/mg protein. The depletion in GSH was not due to inhibition of GR: only FZP caused a significant decrease in macrophage GR activity (28.0 ± 5.9 compared with 42.1 ± 8.0 nmol/ml/min in control cells, p < 0.05 by ANOVA followed by Dunnett's test, n = 3). The marked depletion of macrophage GSH indicates a potential toxic interaction in mammalian cells. The new BNZ analogues were rapidly metabolised by hepatocytes, producing N-dealkylated and multiple hydroxylated phase I metabolites, followed by glucuronidation. This rapid metabolism may limit the therapeutic effect of BNZ 1 and 2 if their metabolites are inactive against leishmaniasis and suggest the need to optimise these lead structures further to obtain compounds with reduced rates and extent of metabolism

Impact of a large wildfire on water-soluble organic aerosol in a major urban area: the 2009 Station Fire in Los Angeles County

This study examines the nature of water-soluble organic aerosol measured in Pasadena, CA, under typical conditions and under the influence of a large wildfire (the 2009 Station Fire). During non-fire periods, water-soluble organic carbon (WSOC) variability was driven by photochemical production processes and sea breeze transport, resulting in an average diurnal cycle with a maximum at 15:00 local time (up to 4.9 μg C m^(−3)). During the Station Fire, primary production was a key formation mechanism for WSOC. High concentrations of WSOC (up to 41 μg C m^(−3)) in smoke plumes advected to the site in the morning hours were tightly correlated with nitrate and chloride, numerous aerosol mass spectrometer (AMS) organic mass spectral markers, and total non-refractory organic mass. Processed residual smoke was transported to the measurement site by the sea breeze later in the day, leading to higher afternoon WSOC levels than on non-fire days. Parameters representing higher degrees of oxidation of organics, including the ratios of the organic metrics m/z 44:m/z 57 and m/z 44:m/z 43, were elevated in those air masses. Intercomparisons of relative amounts of WSOC, organics, m/z 44, and m/z 43 show that the fraction of WSOC comprising acid-oxygenates increased as a function of photochemical aging owing to the conversion of aliphatic and non-acid oxygenated organics to more acid-like organics. The contribution of water-soluble organic species to the organic mass budget (10th–90th percentile values) ranged between 27 %–72 % and 27 %–68 % during fire and non-fire periods, respectively. The seasonal incidence of wildfires in the Los Angeles Basin greatly enhances the importance of water-soluble organics, which has implications for the radiative and hygroscopic properties of the regional aerosol

Quantum transfer operators and quantum scattering

These notes describe a new method to investigate the spectral properties if quantum scattering Hamiltonians, developed in collaboration with J. Sj\"ostrand and M.Zworski. This method consists in constructing a family of "quantized transfer operators" $\{M(z,h)\}$ associated with a classical Poincar\'e section near some fixed classical energy E. These operators are finite dimensional, and have the structure of "open quantum maps". In the semiclassical limit, the family $\{M(z,h)\}$ encode the quantum dynamics near the energy E. In particular, the quantum resonances of the form $E+z$, for $z=O(h)$, are obtained as the roots of $\det(1-M(z,h))=0$.Comment: 18 pages, 3 figure